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Turing Machines, Transition Systems, and Interaction

Turing Machines, Transition Systems, and Interaction. Dina Goldin, U.Connecticut Scott Smolka, SUNY at Stony Brook Peter Wegner, Brown University. computation : finite transformation of input to output input : finite-size (string or number)

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Turing Machines, Transition Systems, and Interaction

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  1. Turing Machines,Transition Systems,and Interaction Dina Goldin, U.ConnecticutScott Smolka, SUNY at Stony BrookPeter Wegner, Brown University EXPRESS'01

  2. computation: finite transformation of input to output input: finite-size (string or number) closed system: all input available at start, all output generated at end Church-Turing thesis: captures this notion of computation computation: ongoing process which performs a task or delivers a service dynamically generated stream of input tokens (requests, percepts, messages) later inputs depend on earlier outputs (lack of modularity) and vice versa (history dependence) objects, processes, components,control devices, reactive systems, intelligent agents Algorithmic vs. Interactive Computation EXPRESS'01

  3. Summary • Persistent Turing Machines (PTMs)an interactive extension of the TM model • Interactive Transition Systems (ITSs)effective transition systems induced by PTMs • Unbounded non-determinismexhibited by ITSs • It pays to be persistentexpressiveness of persistent vs. amnesic computation EXPRESS'01

  4. input work S output Nondeterministic 3-tape TMs • Configurations: s - current state w1- contents of input tape w2- contents of work tape w3- contents of output tape n1 , n2 , n3- tape head posns • Computation is a sequence of transitions: EXPRESS'01

  5. ® | < s0, win, w, e, 1, 1, 1 > < sh, win, w’, wout, 1, 1, 1 > N3TM macrosteps win win w w’ So e Sh wout Þ < win, w > < w’, wout > Notation: M EXPRESS'01

  6. Divergent Computation If computation diverges starting in configuration corresponding macrostep notation is: For all winS*, < s0, win, w, e, 1, 1, 1 > Þ < win, w > < sdiv, t > M Þ < win, sdiv > < sdiv, t > M EXPRESS'01

  7. in1 in1 w1 e e out1 S0 Sh Persistent Stream Languages Persistent Turing Machine (PTM): N3TM with persistent stream-based computational semantics • Persistent Stream Language of a PTM: set of streams • Conductive stream semantics: in2 in2 w1 w2 ... e out2 S0 Sh EXPRESS'01

  8. Formal Definition (Coinductive definition, relative to N3TM M and memory w) PSL(M(w)) = { (wi, wo), s’ S | $w’S*: > Ù < w > Þ < w ' , w w , o i s Î ' PSL ( M ( w ' ))} EXPRESS'01

  9. Î PSL ( M ) Latch 1 (1*,1) (1*,1) # (1*,0) (0*,1) (0*,1) 0 (0*,0) PTM Example: • inputs in1; outputs 1 • inputs in2; outputs 1st bit of in1 • inputs in3; outputs 1st bit of in2 • ... • Example: EXPRESS'01

  10. Interactive Transition Systemsover S < S, m, r > • S is set of states • ris initial state (root) • m is transition relation Required to be recursively enumerable EXPRESS'01

  11. ξ(M) = < reach(M), m, e > > Þ < > < >Îm s , w , s ' , w s ' , w < w , s i o o M i From PTMs to ITSs Reachable memories of a PTM M: Set of words (work-tape contents) w encountered after zero or more macrosteps. where iff EXPRESS'01

  12. T1 =bisim T2 if $ an interactive bisim. between them ITS Bisimulation Let be ITSs, i=1,2 is a (strong) interactive bisimulation if: 1. 2. 3. Clause 2. with roles of s and t reversed EXPRESS'01

  13. Theorem: Proof: EXPRESS'01

  14. =1 =2 = ... Infinite Equivalence Hierarchy • Lk(M) = stream prefix language of PTM Mset of prefixes of length  k for streams in PSL(M). • L (M) = UkLk(M) • Corresponding notion of equivalence: M1 =k M2 : Lk(M1) = Lk ( M2 ) EXPRESS'01

  15. =PSL =1 =2 = ... Equivalence Hierarchy Gap • Proof: construct PTMs M1 and M2 where L(M1) = L (M2 )but PSL (M1 ) = PSL (M2 ) • Note: M2 exhibits unbounded non-determinism / EXPRESS'01

  16. Example of Unbounded Nondeterminism MUD ignores inputs, output 0 or 1 with each macrostep. On 1st macrostep, initializes a persistent string n of 1’s: while true do write ‘1’ on the work tape, move head to the right; nondeterministically choose to exit loop or continue The output at every macrostep is determined as follows: if n > 0 then decrement n by 1 and output ‘1’; else output ‘0’ EXPRESS'01

  17. (S*, 1) (S*, 1) (S*, 1) (S*, 1) ITS for MUD sdiv (S*, t) (S*, t) e ... (S*, 1) (S*, 1) (S*, 1) (S*, 1) (S*, 1) (S*, 0) ... n = 0 n = 1 n = 2 n = 3 EXPRESS'01

  18. Amnesic PTM Computation:stream-based but not persistent > Þ < w', wo > Ù < e w , i s Î ' PSL ( M ( w ' ))} EXPRESS'01

  19. in1 in1 e w1 e out1 S0 Sh Amnesic PTM Computation in2 in2 e w2 ... e out2 S0 Sh Example: outi = ini2 PTM M is amnesic if PSL(M)ASL EXPRESS'01

  20. It pays to be Persistent ASL PSL Proof: Given an N3TM M, construct M’such that PSL(M') = ASL(M) Consider 3rd elem. (0,0) of sio for Mlatch!For any M with sio in ASL(M), there will also be a stream in ASL(M) with (0,0) as 1st element.Therefore, for all M, ASL(M) PSL(Mlatch). EXPRESS'01

  21. Summary of Results =ASL = =PSL =1 =2 = ... =ms PTMs ITSs =ISL =bisim =iso EXPRESS'01

  22. Modeling Interactive Computation: Related Work • Reactive and embedded systems • Dataflow, process algebra, I/O automata, synchronous languages, finite/pushdown automata over infinite words, interaction games, online algorithms • Sequential Interaction Machines [Wegner & Goldin] EXPRESS'01

  23. Future Work • Interactive computability • Universal PTM • Interactive complexity • Where are the ports? http://www.cse.uconn.edu/~dqg/papers/ EXPRESS'01

  24. Interactive Computability • A stream languageL is interactively computable if L PSL (properties of L expressed in Temporal Logic) • A behaviorB is interactively computable if B is interaction bisimilar to an ITS TT EXPRESS'01

  25. ITS Isomorphism Let be ITSs, i=1,2 1. 2. EXPRESS'01

  26. Interactive Stream Equivalence • Infinite sequences of input/output token-pairs emanating from a particular ITS state • For an ITS T and state s, ISL(T(s)) [and ISL(T)] are defined similarly to PSL(M(s)) [and PSL(M)] T1 =ISLT2 if ISL(T1) = ISL(T2) EXPRESS'01

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